1. Field of the Invention
Embodiments of the present invention relate to differential scanning calorimetry. More particularly, embodiments of the present invention relate to an apparatus and method for measuring sample heat flow rate in a differential scanning calorimeter.
2. Background of the Invention
Differential scanning calorimetry is an analytical technique in which the rate of heat flow to and from a sample under analysis is measured while the sample is exposed to a dynamic temperature program. The temperature program may be comprised of constant heating or cooling rate segments combined with isothermal segments, all of which may have superposed temperature modulation of various forms. The temperature modulation can be periodic or aperiodic. Differential scanning calorimetry techniques using temperature modulation are referred to by a variety of names including modulated DSC, temperature modulated DSC, and dynamic DSC.
At this time, there are two different types of DSC instruments in commercial production. One type of DSC instrument operates on the principle of compensation of the thermal effect, and is typically referred to as power compensation DSC. The second type of DSC instrument operates on the principal of measurement of a temperature difference, and is typically referred to as heat flux DSC.
In general, DSC instruments that operate on the heat flux principle use the following equation to obtain the heat flow rate from the measured temperature difference:
                              q          .                =                              Δ            ⁢                                                  ⁢            T                                K            ⁡                          (              T              )                                                          (        1        )            where ΔT is the measured temperature difference and K(T) is a temperature dependent proportionality factor that is characteristic of the particular calorimeter in question. Dimensionally, the proportionality factor K(T) must have the units of a thermal resistance, i.e., temperature divided by energy per unit of time, for example), ° C./joule/second, i.e., ° C./watt. The physical significance of the proportionality constant is that it represents the effective thermal resistance of the heat flow path between the sample and the controlled temperature enclosure or surroundings of the calorimeter. In the case of a differential scanning calorimeter, the temperature difference is measured between two nominally identical sample and reference calorimeters where the sample under analysis is placed in a sample container that is installed in the sample calorimeter, and an inert reference material is placed in the reference calorimeter. Despite its near universal application, equation (1) may be shown to be a great over-simplification of the actual sample heat flow rate. See, for example, G. W. H. Höhne, W. Hemminger, H. J. Flammersheim, “Differential Scanning Calorimetry,” Springer, Berlin, 1996, pp. 21-33, which is hereby incorporated by reference herein in its entirety.
As shown in R. L. Danley, Thermochim. Acta 395 (2003) 201 and disclosed in U.S. Pat. No. 6,431,747 to Danley, which are hereby incorporated by reference herein in their entirety, a heat flow rate measurement method has been developed that avoids many of the assumptions implicit in equation (1). However, this heat flow rate measurement requires the measurement of two temperature differences and a calibration method to measure the heat transfer characteristics of the individual sample and reference calorimeters. In that method, the sample and reference heat flow rates are measured separately using the following equations:
                                          q            .                    s                =                                            Δ              ⁢                                                          ⁢                              T                0                                                    R              s                                -                                    C              s                        ⁢                                          T                .                            s                                                          (        2        )                                                      q            .                    r                =                                                            Δ                ⁢                                                                  ⁢                                  T                  0                                            +                              Δ                ⁢                                                                  ⁢                T                                                    R              r                                -                                    C              r                        ⁡                          (                                                                    T                    .                                    s                                -                                  Δ                  ⁢                                                                          ⁢                                      T                    .                                                              )                                                          (        3        )            where ΔT0 is a second temperature difference measured across the thermal resistance of the sample calorimeter, Ts is the temperature of the sample calorimeter, Rs and Rr are the thermal resistances of the sample and reference calorimeters, and Cs and Cr are the heat capacities of the sample and reference calorimeters. The thermal resistances and heat capacities of the calorimeters are determined from the calibration method mentioned above. The difference between the sample and reference calorimeter heat flow rates may be taken, resulting in a sample heat flow rate measurement that includes differences between the thermal resistances and heat capacities of the sample and reference calorimeters and which accounts for heating rate differences between the two calorimeters:
                              q          .                =                              -                                          Δ                ⁢                                                                  ⁢                T                                            R                r                                              +                      Δ            ⁢                                                  ⁢                                          T                0                            ⁡                              (                                                      1                                          R                      s                                                        -                                      1                                          R                      r                                                                      )                                              +                                    (                                                C                  r                                -                                  C                  s                                            )                        ⁢                                          ⅆ                                  T                  s                                                            ⅆ                t                                              -                                    C              r                        ⁢                                                            ⅆ                  Δ                                ⁢                                                                  ⁢                T                                            ⅆ                t                                                                        (        4        )            It should be noted that the first term of heat flow rate equation (4) is essentially the same as equation (1).
Alternatively, if the effects of heating rates and mass differences between the sample and reference containers are taken into account, the following heat flow rate equation may be used:
                              q          .                =                                            q              .                        s                    -                                                    q                .                            r                        ⁡                          (                                                                    m                                          p                      ⁢                                                                                          ⁢                      s                                                        ⁢                                                            T                      .                                                              p                      ⁢                                                                                          ⁢                      s                                                                                                            m                    pr                                    ⁢                                                            T                      .                                        pr                                                              )                                                          (        5        )            where mps and mpr are the masses of the sample and reference containers and Tps and Tpr are the temperatures of the sample and reference containers, which are found from:Tps=Ts−{dot over (q)}sRp  (6)Tpr=Tr−{dot over (q)}rRp  (7)where Ts is the temperature of the sample calorimeter, Tr is the temperature of the reference calorimeter, and Rp is the thermal contact resistance between each calorimeter and its sample container.
The precision of the heat flow rate measurement depends on how repeatable the measured signals are for a given sample heat flow rate. Boersma (see, S. L. Boersma, J. Am. Ceram. Soc. 38 (1955) 281) found that by mounting the temperature sensors within the sample holders rather than within the sample material the precision of a differential thermal analysis apparatus was greatly improved, enabling quantitative heat flow rate measurements. The measured temperature and temperature differences depend upon the position of the temperature sensors relative to the sample and reference and to other components of the calorimeter.
Thermal contact resistance is the impediment to the flow of heat across the interface between two solid surfaces. Thermal resistance is due to form imperfections in the two mating surfaces resulting in limited direct contact between them. Thus, heat exchange between two solid surfaces that nominally contact each other over a given area is by conduction between regions in the given area where the two surfaces are in direct contact, and by conduction through the gas within the interstitial spaces between the regions of direct contact. Convective heat exchange generally does not occur because the interstitial spaces are small and the viscosity of the gas is sufficiently large that it prevents motion of the gas due to temperature differences between the surfaces. Radiation heat exchange occurs between the two contact surfaces but is typically insignificant because the temperature differences are generally quite small.
In DSC, there are two contact resistances that are of concern. The first contact resistance of interest in DSC is that between the sample and its container, as well as the reference material, if used, and its container. The second contact resistance of interest in DSC is that between the containers and the calorimeters Given that the sample size and shape may vary greatly between samples to be analyzed, the thermal contact resistance between the sample and its container may not be easily controlled. Because the form imperfections of any given sensor and container mating surface differ for different containers or sensors, the contact resistance between any container and sensor combination is different from that of another container/sensor combination and also differs when the relative orientation of the container and sensor surfaces is changed for the same container/sensor combination. Thus, the thermal contact resistance between a DSC sensor and the sample and reference containers installed on the sensor changes whenever a new container is introduced, or even when the same sample container is removed from the DSC and subsequently replaced. These changes result in a different distribution of heat flow between the two surfaces and a different temperature distribution within the container and the calorimeter. If the temperature differences and the temperatures of the calorimeter are measured beneath the region of contact between the pan and sensor, their values are affected by the changes in magnitude and distribution of contact resistance between the surfaces.
U.S. Pat. No. 3,554,002 to Harden, et al. (“Harden”), discloses a differential thermal analysis sample cell in which wires are connected to a sample cell assembly constructed of one of a pair of thermocouple materials. Three wires are joined to the cell assembly, one of which is of the same thermoelectric material as the cell, while the other two are of the opposite material of the thermocouple pair. Each of the wires of the opposite material to the cell material are connected in a position that is axisymmetric with respect to each of a sample and a reference location within the cell, the sample and reference having a cylindrical form. The connection of the wire of like material to the cell assembly may be located anywhere on the cell assembly. A differential temperature is measured between the two wires of opposite thermoelectric composition to the cell, and a temperature is measured between the wire of like thermoelectric composition to the cell material and the wire of opposite thermoelectric composition to the cell material that is attached axisymmetrically with respect to the sample position. Thus, the differential temperature is measured between the two points of attachment of wires of opposite thermoelectric material to the cell material. The measured temperature difference depends on the magnitude of the heat flow rates between the sample and reference and the cell and on the position of the sample and reference materials within the cell and the distribution of thermal contact resistance between those materials and the cell. The temperature measured between the wire attached to the axisymmetric position of the sample position and the wire of like thermoelectric composition attached to the sample cell is taken as the sample temperature. Using a differential thermal analysis apparatus of the type disclosed in Harden, sample heat flow rate may be measured using equation (1).
U.S. Pat. No. 4,095,453 to Woo (“Woo”) discloses a heat flux differential scanning calorimeter that uses a planar thermoelectric disk, with a disk type area thermocouple beneath each of the sample and reference positions of the calorimeter. These two thermocouples are connected to measure the differential temperature between the sample and reference positions. The sample heat flow rate is obtained from the instrument using equation (1). The claimed advantage of the apparatus disclosed in Woo is the improved reproducibility of the heat flow rate signal by using these area thermocouples as opposed to point thermocouples as used in Harden. According to Woo, this reduces the effects of variations in the differential temperature measurement resulting from variations in the positions of the sample and reference containers within the apparatus and variations in thermal contact resistance between the apparatus and the sample and reference containers.
U.S. Pat. No. 6,431,747 to Danley (“Danley”) discloses a heat flux differential scanning calorimeter sensor using a disk type thermocouple system similar to that of Woo to improve reproducibility of the heat flow rate signal and that uses the heat flow rate measurements of equations (2) through (7). In addition, the DSC of Danley has the advantage of greatly improving separation of the sample and reference heat flow signals over the apparatus disclosed in Harden and Woo.
In Harden, Woo, and Danley, the differential temperature is measured using a single differential thermocouple. However, the differential temperature may also be measured using a thermopile that consists of a series of thermocouples connected in series and arranged so that every other thermocouple junction is located within one of a sample and a reference region between which the differential temperature is to be measured. The advantage of using a thermopile over using a single thermocouple is that a larger electrical signal may be generated using a thermopile.
However, thermopile sensors are in general more complex and more difficult to manufacture. Moreover, in reality, due to the available choice of thermocouple materials and other design factors, the actual advantages of using thermopiles is generally less than otherwise might be expected. Differences notwithstanding, all of the above comments that have been made with respect to the method of heat flow rate measurement and sample and reference container position and contact resistance apply equally to DSC apparatus using thermopile differential temperature sensing.
U.S. Pat. No. 5,033,866 to Kehl, et al. (“Kehl”), discloses a thermopile thermal analysis sensor that may be used to measure sample heat flow rate using the method of equation (1) in which thermoelectric materials are deposited on a ceramic substrate using thick film processes such as those employed in the manufacture of integrated circuits and hybrid electronic devices. The temperature difference is measured between a circular region beneath the sample position of the sensor and a circular region beneath the reference position of the sensor.
U.S. Pat. No. 5,288,147 to Schaefer et al. (“Schaefer”) discloses a differential thermal analysis sensor that incorporates two thermopiles that may be used to measure sample heat flow rate using either equation (1) or the method of equations (2) through (7). As with the apparatus disclosed in Kehl, the thermopiles are formed using thick film processes. However, unlike Kehl, the apparatus disclosed in Schaefer uses two separate thermopiles that measure the temperature difference between the sample position and a second region of the sensor located within a portion of the sensor assembly that is attached to the DSC enclosure and between the reference position and the second region of the sensor.